Download Equilibrium and Efficiency

Chapter 2
Equilibrium and Efficiency
2.1
Introduction
The link between competition and efficiency can be traced back, at least, to
Adam Smith’s 18th century description of the working of the invisible hand.
Smith’s description of individually-motivated decisions being coordinated to
produce a socially-efficient outcome is a powerful one which has found resonance in policy circles ever since. The expression of the efficiency argument
in the language of formal economics, and the deeper understanding that comes
with it, is a more recent innovation.
The focus of this chapter is to review what is meant by competition and
to describe equilibrium in a competitive economy. The model of competition
combines independent decision-making of consumers and firms into a complete
model of the economy. Equilibrium is shown to be achieved in the economy by
prices adjusting to equate demand and supply. Most importantly, the chapter
employs the competitive model to demonstrate the efficiency theorems.
That equilibrium prices can always be found which simultaneously equate
demand and supply for all goods is surprising. What is even more remarkable
is that the equilibrium so obtained also has properties of efficiency. Why this
is remarkable is that individual households and firms pursue their independent
objectives with no concern other than their own welfare. Even so, the final state
that emerges achieves efficiency solely through the coordinating role played by
prices.
2.2
Economic Models
Prior to starting the analysis it is worth reflecting upon why economists employ
models to make predictions about the effects of economic policies. Essentially,
models are used because of problems of conducting experiments on economic
systems and because the system is too large and complex to analyze in its
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12
CHAPTER 2. EQUILIBRIUM AND EFFICIENCY
entirety. Moreover, formal modelling ensures that arguments have to be logically
consistent with all the underlying assumptions exposed.
The models used, while inevitably being simplifications of the real economy,
are designed to capture the essential aspects of the problem under study. Although many different models will be studied in this book, there are important
common features which apply to all. Most models in public economics specify
the objectives of the individual agents in the economy (such as firms and consumers), and the constraints they face, and then aggregate individual decisions
to arrive at market demand and supply. The equilibrium of the economy is then
determined and, in a policy analysis, the effects of government choice variables
upon this are calculated. This is done with various degrees of detail. Sometimes
only a single market is studied — which is the case of partial equilibrium analysis.
At other times, general equilibrium analysis is used with many markets analyzed
simultaneously. Similarly, the number of firms and consumers varies from one
or two to very many.
An essential consideration in the choice of the level of detail for a model
is that its equilibrium must demonstrate a dependence upon policy that gives
insight into the functioning of the actual economy. If the model is too highly
specified it may not be capable of capturing important forms of response. On
the other hand, if it is too general it may not be able to provide any clear
prediction. The theory described in this book will show how this trade-off
can be successfully resolved. Achieving a successful compromise between these
competing objectives is the “art” of economic modelling.
2.3
Competitive Economies
The essential feature of competition is that the consumers and firms in the economy do not consider their actions to have any effect upon prices. Consequently,
in making decisions they treat the prices they observe in the market place as
fixed (or parametric). This assumption can be justified when all consumers and
firms are truly negligible in size relative to the market. In such a case the quantity traded by an individual consumer or firm is not sufficient to change the
market price. But the assumption that the agents view prices as parametric can
also be imposed as a modelling tool even in an economy with a single consumer
and a single firm.
This defining characteristic of competition places a focus upon the role of
prices which is maintained throughout the chapter. Prices measure values and
are the signals which guide the decisions of firms and consumers. It was the
exploration of what determined the relative values of different goods and services
that lead to the formulation of the competitive model. The adjustment of prices
equates supply and demand to ensure that equilibrium is achieved. The role
of prices in coordinating the decisions of independent economic agents is also
crucial for the attainment of economic efficiency.
The secondary feature of the economies in this chapter is that all agents have
access to the same information or — in formal terminology — that information is
2.3. COMPETITIVE ECONOMIES
13
symmetric. This does not imply that there cannot be uncertainty but only that
when there is uncertainty all agents are equally uninformed. Put differently, no
agent is permitted to have an informational advantage. As an example, this
assumption allows the future profit levels of firms to be uncertain and for shares
in the firms to be traded on the basis of individual assessments of future profits.
What it does not allow is for the directors of the firms to be better informed
than other shareholders about future prospects and to trade profitably on the
basis of this information advantage.
Two forms of the competitive model are introduced in this chapter. The first
form is an exchange economy in which there is no production. Initial stocks of
goods are held by consumers and economic activity occurs through the trade
of these stocks to mutual advantage. The second form of competitive economy introduces production. This is undertaken by firms with given production
technologies who use inputs to produce outputs and distribute their profits as
dividends to consumers.
2.3.1
The Exchange Economy
The exchange economy models the simplest form of economic activity: the trade
of commodities between two parties in order to obtain mutual advantage. Despite the simplicity of this model, it is a surprisingly instructive tool for obtaining
fundamental insights about taxation and tax policy. This will become evident as
we proceed. This section will present a description of a two-consumer, two-good
exchange economy. The restriction on the number of goods and consumers does
not alter any of the conclusions that will be derived — they will all extend to
larger numbers. What restricting the numbers does is allow the economy to be
displayed and analyzed in a simple diagram.
Each of the two consumers has an initial stock, or endowment, of the economy’s two goods. The endowments can be interpreted literally as stocks of
goods, or less literally as human capital, and are the quantities that are available
for trade. Given the absence of production, these quantities remains constant.
The consumers exchange quantities of the two commodities in order to achieve
consumption plans that are preferred to their initial endowments. The rate at
which one commodity can be exchanged for the other is given by the market
prices. Both consumers believe that their behavior cannot affect these prices.
This is the fundamental assumption of competitive price-taking behavior. More
will be said about the validity and interpretation of this in Section 2.6.
A consumer is described by their endowments
and their preferences. The
endowment of consumer h is denoted by ω h = ω h1 , ω h2 , where ωhi ≥ 0 is h’s
initial stock of good
prices are p1 and p2 a consumption plan for
i. When
consumer h, xh = xh1 , xh2 , is affordable if it satisfies the budget constraint
p1 xh1 + p2 xh2 = p1 ω h1 + p2 ω h2 .
(2.1)
The preferences of each consumer are described by their utility function. This
function should be seen as a representation of the consumer’s indifference curves
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CHAPTER 2. EQUILIBRIUM AND EFFICIENCY
and does not imply any comparability of utility levels between consumers —
the issue of comparability is taken up in Chapter 12. The utility function for
consumer h is denoted by
U h = U h xh1 , xh2 .
(2.2)
It is assumed that the consumers enjoy the goods (so the marginal utility of
consumption is positive for both goods) and that the indifference curves have
the standard convex shape.
This economy can be pictured in a simple diagram which allows the role of
prices in achieving equilibrium to be explored. The diagram is constructed by
noting that the total consumption of the two consumers must equal the available
stock of the goods, where the stock is determined by the endowments. Any pair
of consumption plans that satisfies this requirement is called a feasible plan for
the economy. A plan for the economy is feasible if the consumption levels can
be met from the endowments, so
x1i + x2i = ω1i + ω 2i , i = 1, 2.
(2.3)
The consumption plans satisfying (2.3) can be represented as points in a rectangle with sides of length ω 11 + ω 21 and ω 12 + ω 22 . In this rectangle the south-west
corner can be treated as the zero consumption point for consumer 1 and the
north-east corner as the zero consumption point for consumer 2. The consumption of good 1 for consumer 1 is then measured horizontally from the south-west
corner and for consumer 2 horizontally from the north-east corner. Measurements for good 2 are made vertically.
The diagram constructed in this way is called an Edgeworth box and a typical
box is shown in Figure 2.1. It should be noted that the method of construction
results in the endowment point, marked ω, being the initial endowment point
for both consumers.
The Edgeworth box is completed by adding the preferences and budget constraints of the consumers. The indifference curves of consumer 1 are drawn
relative to the south-west corner and those of consumer 2 relative to the northeast corner. From (2.1), it can be seen that the budget constraint for both
consumers must pass through the endowment point since they can always afford their endowment. The endowment point is common to both consumers, so
a single budget line through the endowment point with gradient − pp12 captures
the market opportunities of the two consumers. Thus, viewed from the southwest it is the budget line of consumer 1 and viewed from the north-east the
budget line of consumer 2. Given the budget line determined by the prices p1
and p2 , the utility-maximizing choices for the two consumers are characterized
by the standard tangency condition between the highest attainable indifference
curve and the budget line. This is illustrated in Figure 2.2, where x1 denotes
the choice of consumer 1 and x2 that of 2.
In an equilibrium of the economy, supply is equal to demand. This is assumed
to be achieved via the adjustment of prices. The prices at which supply is equal
to demand are called equilibrium prices. How such prices are arrived at will
2.3. COMPETITIVE ECONOMIES
15
x12
ω 12 + ω 22
x12
•
2
x
x22
•
1
ω
x11
ω11 + ω12
Figure 2.1: An Edgeworth Box
2
x2
x1
ω
1
Figure 2.2: Preferences and Demand
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CHAPTER 2. EQUILIBRIUM AND EFFICIENCY
2
x2
x1
ω
1
Figure 2.3: Relative Price Change
be discussed later. For the present, the focus will be placed on the nature of
equilibrium and its properties. The consumer choices shown in Figure 2.2 do
not constitute an equilibrium for the economy. This can be seen by summing
the demands and comparing these to the level of the endowments. Doing this
shows that the demand for good 1 exceeds the endowment but the demand for
good 2 falls short. To achieve an equilibrium position, the relative prices of
the goods must change. An increase in the relative price of good 1 raises the
absolute value of the gradient − pp12 of the budget line, making the budget line
steeper. It becomes flatter if the relative price of good 1 falls. At all prices, it
continues to pass through the endowment point so a change in relative prices
sees the budget line pivot about the endowment point.
The effect of a relative price change upon the budget constraint is shown
in Figure 2.3. In the figure the price of good 1 has increased relative to the
price of good 2. This causes the budget constraint to pivot upwards around
the endowment point. As a consequence of this change the consumers will now
select consumption plans on this new budget constraint.
The dependence of the consumption levels upon prices is summarized in the
consumers’ demand functions. Taking the prices as given, the consumers choose
their consumption plans to reach the highest attainable utility level subject to
their budget constraints. The level of demand for good i from consumer h is
xhi = xhi (p1 , p2 ) . Using the demand functions, demand is equal to supply for
the economy when the prices are such that
x1i (p1 , p2 ) + x2i (p1 , p2 ) = ω 1i + ω2i , i = 1, 2.
(2.4)
Study of the Edgeworth box shows that such an equilibrium is achieved when the
prices lead to a budget line on which the indifference curves of the consumers
have a point of common tangency. Such an equilibrium is shown in Figure
2.4. Having illustrated an equilibrium, the question is raised of whether an
2.3. COMPETITIVE ECONOMIES
17
2
x2
x
1
ω
1
Figure 2.4: Equilibrium
equilibrium is guaranteed to exist. In fact, under reasonable assumptions, it
will always do so. More important for public economics is the issue of whether
the equilibrium has any desirable features from a welfare perspective. This is
discussed in depth in the Section 2.4 in which the Edgeworth box sees substantial
use.
Two further points now need to be made that are important for understanding the functioning of the model. These concern the number of prices that can
be determined and the number of independent equilibrium equations. In the
equilibrium conditions (2.4) there are two equations to be satisfied by the two
equilibrium prices. It is now argued that the model can determine only the ratio
of prices and not the actual prices. Accepting this, we would seem to be in a
position where there is one price ratio attempting to solve two equations. If this
were the case, a solution would be unlikely and we would be in the position of
having a model that generally did not have an equilibrium. This situation is
resolved by noting that there is a relationship between the two equilibrium conditions which ensures that there is only one independent equation. The single
price ratio then has to solve a single equation, thus making it possible for there
to be always a solution.
The first point is developed by observing that the budget constraint always
passes through the endowment point and its gradient is determined by the price
ratio. The consequence of this is that only the value of p1 relative to p2 matters
in determining demands and supplies rather than the absolute values. The
economic explanation for this fact is that consumers are only concerned with
the real purchasing power embodied in their endowment, and not with the level
of prices. Since their nominal income is equal to the value of the endowment,
any change in the level of prices raises nominal income just as much as it raises
the cost of purchases. This leaves real incomes unchanged.
The fact that only relative prices matter is also reflected in the demand
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CHAPTER 2. EQUILIBRIUM AND EFFICIENCY
functions. If xhi (p1 , p2 ) is the level of demand at prices p1 and p2 , then it must
be that case that xhi (p1 , p2 ) = xhi (λp1 , λp2 ) , for λ > 0. A demand function
having this property is said to be homogenous of degree 0. In terms of what can
be learnt from the model, the homogeneity shows that only relative prices can be
determined at equilibrium not the level of prices. So, given a set of equilibrium
prices any scaling up, or down, of these will also be equilibrium prices since
the change will not alter the level of demand. This is as it should be, since
all that matters for the consumers is the rate at which they can exchange one
commodity for another, and this is measured by the relative prices. This can
be seen in the Edgeworth box. The budget constraint always goes through the
endowment point so only its gradient can change and this is determined by the
relative prices.
In order to analyze the model the indeterminacy of the level of prices needs
to be removed. This is achieved by adopting a price normalization which is
simply a method of fixing a scale for prices. There are numerous ways to do
this. The simplest way is to select a commodity as numeraire, which means
that its price is fixed at one, and measure all other prices relative to this. The
numeraire chosen in this way can be thought of as the unit of account for the
economy. This is the role usually played by money but, formally, there is no
money in this economy.
The second point is to demonstrate the dependence between the two equilibrium equations. It can be seen that at the disequilibrium position shown in
Figure 2.2 the demand for good 1 exceeds its supply whereas the supply of good
2 exceeds demand. Considering other budget lines and indifference curves in
the Edgeworth box will show that whenever there is an excess of demand for
one good there is a corresponding deficit of demand for the other. There is, in
fact, a very precise relationship between the excess and the deficit which can
be captured in the following way. The level of excess demand for good i is the
difference between demand and supply and is defined by Zi = x1i + x2i − ω 1i − ω 2i .
Using the definition the value of excess demand can be calculated as
p1 Z1 + p2 Z2
=
2
i=1
pi x1i + x2i − ω1i − ω 2i
2
=
p1 xh1 + p2 xh2 − p1 ω h1 − p2 ω h2
h=1
= 0,
(2.5)
where the second equality is a consequence of the budget constraints in (2.1).
The relationship in (2.5) is known as Walras’ Law and states that the value
of excess demand is zero. This must hold for any set of prices so it provides a
connection between the extent of disequilibrium and prices. In essence, Walras’
Law is simply an aggregate budget constraint for the economy. Since all consumers are equating their expenditure to their income, so must the economy as
a whole.
Walras’ Law implies that the equilibrium equations are interdependent. Since
2.3. COMPETITIVE ECONOMIES
19
Z 2 (1, p 2 )
p2
Figure 2.5: Equilibrium and Excess Demand
p1 Z1 + p2 Z2 = 0, if Z1 = 0 then Z2 = 0 (and vice versa). That is, if demand
is equal to supply for good 1 then demand must also equal supply for good 2.
Equilibrium in one market necessarily implies equilibrium in the other. This observation allows the construction of a simple diagram to illustrate equilibrium.
Choose good 1 as the numeraire (so p1 = 1) and plot the excess demand for
good 2 as a function of p2 . The equilibrium for the economy is then found where
the graph of excess demand crosses the horizontal axis. At this point, excess
demand for good 2 is zero so, by Walras’ Law, it must also be zero for good
1. An excess demand function is illustrated in Figure 2.5 for an economy that
has 3 equilibria. This excess demand function demonstrates why at least one
equilibrium will exist. As p2 falls towards zero then demand will exceed supply
(good 2 becomes increasingly attractive to purchase) making excess demand
positive. Conversely, as the price of good 2 rises it will become increasingly
attractive to sell, hence resulting in a negative value of excess demand for high
values of p2 . Since excess demand is positive for small values of p2 and negative
for high values, there must be at least one point in between where it is zero.
Finally, it should be noted that the arguments made above can be extended
to include additional consumers and additional goods. Income, in terms of an
endowment of many goods, and expenditure, defined in the same way, must
remain equal for each consumer. The demand functions that result from the
maximization of utility are homogenous of degree zero in prices. Walras’ Law
continues to hold so the value of excess demand remains zero. The number of
price ratios and the number of independent equilibrium conditions are always
one less than the number of goods.
2.3.2
Production and Exchange
The addition of production to the exchange economy provides a complete model
of economic activity. Such an economy allows a wealth of detail to be included.
Some goods can be present as initial endowments (such as labor), others can be
consumption goods produced from the initial endowments, while some goods,
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CHAPTER 2. EQUILIBRIUM AND EFFICIENCY
Good 2
Y
j
3
-2
Good 1
Figure 2.6: A Typical Production Set
intermediates, can be produced by one productive process and used as inputs
into another. The fully-developed model of the competition is called the ArrowDebreu economy in honor of its original constructors.
An economy with production consists of consumers (or households) and producers (or firms). The firms use inputs to produce outputs with the intention
of maximizing their profits. Each firm has available a production technology
which describes the ways in which it can use inputs to produce outputs. The
consumers have preferences and initial endowments as they did in the exchange
economy, but they now also hold shares in the firms. The firms’ profits are distributed as dividends in proportion to the shareholdings. The consumers receive
income from the sale of their initial endowments (notably their labor time) and
from the dividend payments.
Each firm is characterized by its production set which summarizes the production technology it has available. A production technology can be thought of
as a complete list of ways that the firm can turn inputs into outputs. In other
words, it catalogs all the production methods of which the firm has knowledge.
For firm j operating in an economy with two goods a typical production set,
denoted Y j , is illustrated in Figure 2.6. This figure employs the standard convention of measuring inputs as negative numbers and outputs as positive. The
reason for adopting this convention is that the use of a unit of a good as an input
represent a subtraction from the stock of that good available for consumption
Consider the firm shown in Figure 2.6 choosing the production plan y1j =
−2, y2j = 3. When faced with prices p1 = 2, p2 = 2, the firm’s profit is
πj = p1 y1j + p2 y2j = 2 × (−2) + 2 × 3 = 2.
(2.6)
The positive part of this sum can be given the interpretation of sales revenue
and the negative part that of production costs. This is equivalent to writing
profit as the difference between revenue and cost. Written in this way, (2.6)
gives a simple expression of the relation between prices and production choices.
2.3. COMPETITIVE ECONOMIES
π =0
21
π >0
Good 2
π <0
y*
p2
p1
Good 1
Figure 2.7: Profit Maximization
The process of profit maximization is illustrated in Figure 2.7. Under the
competitive assumption the firm takes the prices p1 and p2 as given. These
prices are used to construct isoprofit curves, which show all production plans
that give a specific level of profit. For example, all the production plans on
the isoprofit curve labelled π = 0 will lead to a profit level of 0. Production
plans on higher isoprofit curves lead to progressively larger profits and those
on lower curves to negative profits. Since doing nothing (which means choosing
y1j = y2j = 0) earns zero profit, the π = 0 isoprofit curve always passes through
the origin.
The profit-maximizing firm will choose a production plan that places it upon
the highest attainable isoprofit curve. What restricts the choice is the technology
that is available as described by the production set. In Figure 2.7 the production
plan that maximizes profit is shown by y ∗ which is located at a point of tangency
between the highest attainable isoprofit curve and the production set. There is
no other technologically-feasible plan which can attain higher profit.
It should be noted how the isoprofit curves are determined by the prices. In
fact, the geometry is that the isoprofit curves are at right-angles to the price
vector. The angle of the price vector is determined by the price ratio, pp21 , so that
a change in relative prices will alter the gradient of the isoprofit curves. The
figure can be used to predict the effect of relative price changes. For instance,
if p1 increases relative to p2 , which can be interpreted as the price of the input
(good 1) rising in comparison to the price of the output (good 2), the price
vector become flatter. This makes the isoprofit curves steeper, so the optimal
choice must move round the boundary of the production set towards the origin.
The use of the input and the production of the output both fall.
Now consider an economy with n goods. The price of good i is denoted
pi . Production is carried out by m firms. Each firm uses inputs to produce
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CHAPTER 2. EQUILIBRIUM AND EFFICIENCY
outputs and maximizes profits given the market prices. Demand comes from
the H consumers in the economy. They aim to maximize their utility. The total
supply of each good is the sum of the production of it by firms and the initial
endowment of it held by the consumers.
Each firm chooses a production plan y j = y1j , ..., ynj . This production plan
is chosen to maximize profits subject to the constraint that the chosen plan must
be in the production set. From this maximization can be determined firm j’s
supply function for good i as yij = yij (p), where p = (p1 , ..., pn ). The level of
n
profit is πj = i=1 pi yij (p) = πj (p) , which also depends upon prices.
Aggregate supply from the production sector of the economy is obtained
from the individual supply decisions of the individual firms by summing across
the firms. This gives the aggregate supply of good i as
Yi (p) =
m
yij (p) .
(2.7)
j=1
Since some goods must be inputs, and others outputs, aggregate supply can be
positive (the total activity of the firms adds to the stock of the good) or negative
(the total activity of the firms subtracts from the stock).
Each consumer has an initial endowment of commodities and also a set of
shareholdings in firms. The latter assumption makes this a private ownership
economy in which the means of production are ultimately owned by individuals.
In the present version of the model, these shareholdings are exogenously given
and remain fixed. A more-developed version would introduce a stock market
and allow them to be traded. For consumer h the initial endowment is denoted
ω h and the shareholding in firm j is θ hj . The firms must be fully owned by the
h
consumers so H
h=1 θj = 1. That is, the shares in the firms must sum to one.
Consumer h chooses a consumption plan xh to maximize utility subject to the
budget constraint
n
n
m
h
h
pi xi =
pi ω i +
θhj π j .
(2.8)
i=1
i=1
j=1
This budget constraint requires that the value of expenditure is not more than
the value of the endowment plus income received as dividends from firms. Since
firms always have the option of going out of business (and hence earning zero
profit), dividend income must be non-negative. The profit level of each firm
is dependent upon prices. A change in prices therefore affects a consumer’s
budget constraint through a change in the value of their endowment and through
a change in dividend income. The maximization of utility by the consumer
results in demand for good i from consumer h of the form xhi = xhi (p) . The
level of aggregate demand is found by summing the individual demands of the
consumers to give
H
Xi (p) =
xhi (p) .
(2.9)
h=1
2.4. EFFICIENCY OF COMPETITION
23
The same notion of equilibrium that was used for the exchange economy
can be applied in this economy with production. That is, equilibrium occurs
when supply is equal to demand. The distinction between the two is that supply, which was fixed in the exchange economy, is now variable and dependent
upon the production decisions of firms. Although this adds a further dimension
to the question of the existence of equilibrium, the basic argument why such
an equilibrium always exists is essentially the same as that for the exchange
economy.
As already noted, the equilibrium of the economy occurs when demand is
equal to supply or, equivalently, when excess demand is zero. Excess demand
for good i, Zi (p), can be defined by
Zi (p) = Xi (p) − Yi (p) −
H
ω hi .
(2.10)
h=1
Here excess demand is the difference between demand and the sum of initial
endowment and firms’ supply. The equilibrium occurs when Zi (p) = 0 for all
of the goods i = 1, ..., n. There are standard theorems that prove such an
equilibrium must exist under fairly weak conditions.
The properties established for the exchange economy also apply to this economy with production. Demand is determined only by relative prices (so it is
homogenous of degree zero). Supply is also determined by relative prices. Together, these imply that excess demand is homogenous of degree zero. To determine the equilibrium prices that equate supply to demand, a normalization
must again be used. Typically, one of the goods will be chosen as numeraire
and its price set to one. Equilibrium prices are then those which equate excess
demand to zero.
2.4
Efficiency of Competition
Economics is often defined as the study of scarcity. This viewpoint is reflected
in the concern with the efficient use of resources that runs throughout the core
of the subject. Efficiency would seem to be a simple concept to characterize: if
more cannot be achieved then the outcome is efficient. This is certainly the case
when an individual decision-maker is considered. The individual will employ
their resources to maximize utility subject to the constraints they face. When
utility is maximized, the efficient outcome has been achieved.
Problems arise when there is more than one decision-maker. To be unambiguous about efficiency it is necessary to resolve the potentially competing
needs of different decision-makers. This requires efficiency to be defined with
respect to a set of aggregate preferences. Methods of progressing from individual to aggregate preferences will be discussed in Chapters 10 and 12. The
conclusions obtained there are that the determination of aggregate preferences
is not a simple task. There are two routes we can use to navigate around this
difficulty. The first is to look at a single-consumer economy so that there is
24
CHAPTER 2. EQUILIBRIUM AND EFFICIENCY
no conflict between competing preferences. But with more than one consumer
some creativity has to be used to describe efficiency. The second route is met in
Section 2.4.2 where the concept of Pareto-efficiency is introduced. The trouble
with such creativity is that it leaves the definition of efficiency open to debate.
We will postpone further discussion of this until Chapter 12.
Before proceeding some definitions are needed. A first-best outcome is
achieved when only the production technology and the limited endowments restrict the choice of the decision-maker. Essentially, the first-best is what would
be chosen by an omniscient planner with complete command over resources.
A second-best outcome arises whenever constraints other than technology and
resources are placed upon what the planner can do. Such constraints could be
limits on income redistribution, an inability to remove monopoly power or a
lack of information.
2.4.1
Single Consumer
With a single consumer there is no doubt as to what is good and bad from a
social perspective: the single individual’s preferences can be taken as the social
preferences. To do otherwise would be to deny the validity of the consumer’s
judgements. Hence, if the individual prefers one outcome to another, then
so must society. The unambiguous nature of preferences provides significant
simplification of the discussion of efficiency in the single-consumer economy. In
this case the “best” outcome must be first-best since no constraints on policy
choices have been invoked nor is there an issue of income distribution to consider.
If there is a single firm and a single consumer, the economy with production
can be illustrated in a helpful diagram. This is constructed by superimposing
the profit-maximization diagram for the firm over the choice diagram for the
consumer. Such a model is often called the Robinson Crusoe economy. The
interpretation is that Robinson acts as a firm carrying out production and as a
consumer of the product of the firm. It is then possible to think of Robinson
as a social planner who can coordinate the activities of the firm and producer.
It is also possible (though in this case less compelling!) to think of Robinson
as having a split personality and acting as a profit-maximizing firm on one side
of the market and as a utility-maximizing consumer on the other. In the latter
interpretation, the two sides of Robinson’s personality are reconciled through
the prices on the competitive markets. The important fact is that these two
interpretations lead to exactly the same levels of production and consumption.
The budget constraint of the consumer needs to include the dividend received
from the firm. With two goods, the budget constraint is
p1 [x1 − ω 1 ] + p2 [x2 − ω2 ] = π,
(2.11)
p1 x
1 + p2 x
2 = π,
(2.12)
or
where x
i , the change from the endowment point, is the net consumption of good
i. This is illustrated in Figure 2.8 with good 2 chosen as numeraire. The budget
2.4. EFFICIENCY OF COMPETITION
25
Good 2
~
x*
1
π
p1
Good 1
Figure 2.8: Utility Maximization
constraint (2.12) is always at a right-angle to the price vector and is displaced
above the origin by the value of profit. Utility maximization occurs where the
highest indifference curve is reached given the budget constraint. This results
in net consumption plan x
∗ .
The equilibrium for the economy is shown in Figure 2.9 which superimposes
Figure 2.7 onto 2.8. At the equilibrium, the net consumption plan from the
consumer must match the supply from the firm. The feature that makes this
diagram work is the fact that the consumer receives the entire profit of the firm
so the budget constraint and the isoprofit curve are one and the same. The
height above the origin of both is the level of profit earned by the firm and
received by the consumer. Equilibrium can only arise when the point on the
economy’s production set which equates to profit maximization is the same as
that of utility maximization. This is point x
∗ = y ∗ in Figure 2.9.
It should be noted that the equilibrium is on the boundary of the production
set so that it is efficient: it is not possible for a better outcome to be found in
which more is produced with the same level of input. This captures the efficiency
of production at the competitive equilibrium, about which much more is said
soon. The equilibrium is also the first-best outcome for the single-consumer
economy since it achieves the highest indifference curve possible subject to the
restriction that it is feasible under the technology. This is illustrated in Figure
2.9 where x
∗ is the net level of consumption relative to the endowment point in
the first-best and at the competitive equilibrium.
A simple characterization of this first-best allocation can be given by using
the fact that it is at a tangency point between two curves. The gradient of
the indifference curve is equal to the ratio of the marginal utilities of the two
goods and is called the marginal rate of substitution. This measures the rate
26
CHAPTER 2. EQUILIBRIUM AND EFFICIENCY
Good 2
~
x* = y *
1
π
p1
Good 1
Figure 2.9: Efficient Equilibrium
at which good 1 can be traded for good 2 while maintaining constant utility.
Using subscripts to denote the marginal utilities of the two goods, the marginal
1
rate of substitution is given by M RS1,2 = U
U2 . Similarly, the gradient of the
production possibility set is termed the marginal rate of transformation and
denoted MRT1,2 . The MRT1,2 measures the rate at which good 1 has to be
given up to allow an increase in production of good 2. At the tangency point,
the two gradients are equal so
M RS1,2 = M RT1,2 .
(2.13)
The reason why this equality characterizes the first-best equilibrium can be
explained as follows: The MRS captures the marginal value of good 1 to the
consumer relative to the marginal value of good 2 while the M RT measures the
marginal cost of good 1 relative to the marginal cost of good 2. The first-best
is achieved when the marginal value is equal to the marginal cost.
The market achieves efficiency through the coordinating role of prices. The
consumer maximizes utility subject to their budget constraint. The optimal
choice occurs when the budget constraint is tangential to highest attainable
indifference curve. The condition describing this is that ratio of marginal utilities
is equal to the ratio of prices. Expressed in terms of the M RS this is
p1
MRS1,2 = .
(2.14)
p2
Similarly, profit maximization by the firm occurs when the production possibility
set is tangential to the highest isoprofit curve. Using the M RT , the profitmaximization condition is
p1
(2.15)
MRT1,2 = .
p2
2.4. EFFICIENCY OF COMPETITION
27
Good 2
π =0
~
x* = y *
1
p1
Good 1
Figure 2.10: Constant Returns to Scale
Combining these conditions the competitive equilibrium satisfies
M RS1,2 =
p1
= MRT1,2 .
p2
(2.16)
The condition in (2.16) demonstrates that the competitive equilibrium satisfies
the same condition as the first-best and reveals the essential role of prices. Under
the competitive assumption, both the consumer and the producer are guided
in their decisions by the same price ratio. Each optimizes relative to the price
ratio, hence their decisions are mutually efficient.
There is one special case that is worth noting before moving on. When the
firm has constant returns to scale the efficient production frontier is a straight
line through the origin. The only equilibrium can be when the firm makes zero
profits. If profit was positive at some output level, then the constant returns to
scale allows the firm to double profit by doubling output. Since this argument
could then be repeated, there is no limit to the profit that the firm could make.
Hence the claim that equilibrium profit must be zero. Now the isoprofit curve at
zero profit is also a straight line through the origin. The zero-profit equilibrium
can only arise when this is coincident with the efficient production frontier. At
this equilibrium the price vector is at right angles to both the isoprofit curve
and the production frontier. This is illustrated in Figure 2.10.
There are two further implications of constant returns. Firstly, the equilibrium price ratio is determined by the zero profit condition alone and is independent of demand. Secondly, the profit income of the consumer is zero so the
consumer’s budget constraint also passes through the origin. As this is determined by the same prices as the isoprofit curve, the budget constraint must also
be coincident with the production frontier.
28
CHAPTER 2. EQUILIBRIUM AND EFFICIENCY
In this single-consumer context the equilibrium reached by the market simply
cannot be bettered. Such a strong statement cannot be made when further
consumers are introduced since issues of distribution between consumers then
arise. However, what will remain is the finding that the competitive market
ensures that firms produce at an efficient point on the frontier of the production
set and that the chosen production plan is what is demanded at the equilibrium
prices by the consumer. The key to this coordination are the prices that provide
the signals guiding choices.
2.4.2
Pareto-Efficiency
When there is more than one consumer the simple analysis of the Robinson
Crusoe economy does not apply. Since consumers can have differing views about
the success of an allocation, there is no single, simple measure of efficiency. The
essence of the problem is that of judging between allocations with different
distributional properties. What is needed is some process that can take account
of the potentially diverse views of the consumers and separate efficiency from
distribution.
To achieve this, economists employ the concept of Pareto-efficiency. The
philosophy behind this concept is to interpret efficiency as meaning that there
must be no unexploited economic gains. Testing the efficiency of an allocation
then involves checking whether there are any such gains available. More specifically, Pareto-efficiency judges an allocation by considering whether it is possible
to undertake a reallocation of resources that can benefit at least one consumer
without harming any other. If it were possible to do so, then there would exist
unexploited gains. When no improving reallocation can be found, then the initial position is deemed to be Pareto-efficient. An allocation that satisfies this
test can be viewed as having achieved an efficient distribution of resources. For
the present chapter this concept will be used uncritically. The interpretations
and limitations of this form of efficiency will be discussed in Chapter 12.
To provide a precise statement of Pareto-efficiency that applies in a competitive economy it is first necessary to extend the idea of feasible allocations of
resources that was used in (2.3) to define the Edgeworth box. When production
is included, an allocation of consumption is feasible if it can be produced given
the economy’s initial endowments and production technology. Given the initial
endowment, ω, the consumption allocation x is feasible if there is production
plan y such that
x = y + ω.
(2.17)
Pareto-efficiency is then tested using the feasible allocations. A precise definition
follows.
Definition 1 A feasible consumption allocation x
is Pareto-efficient if there
does not exist an alternative feasible allocation x such that:
;
(i) Allocation x gives all consumers at least as much utility as x
(ii) Allocation x gives at least one consumer more utility than x
.
2.4. EFFICIENCY OF COMPETITION
29
These two conditions can be summarized as saying that allocation x
is
Pareto-efficient if there is no alternative allocation (a move from x
to x) that
can make someone better-off without making anyone worse-off. It is this idea of
being able to make someone better-off without making someone else worse-off
that represents the unexploited economic gains in an inefficient position.
It should be noted even at this stage how Pareto-efficiency is defined by
the negative property of being unable to find anything better than the allocation. This is somewhat different to a definition of efficiency that looks for
some positive property of the allocation. Pareto-efficiency also sidesteps issues
of distribution rather than confronting them. This is why it works with many
consumers. More will be said about this in Chapter 12 when the construction
of social welfare indicators is discussed.
2.4.3
Efficiency in an Exchange Economy
The welfare properties of the economy, which are commonly known as the Two
Theorems of Welfare Economics, are the basis for claims concerning the desirability of the competitive outcome. In brief, the First Theorem states that a
competitive equilibrium is Pareto-efficient and the Second Theorem that any
Pareto-efficient allocation can be decentralized as a competitive equilibrium.
Taken together, they have significant implications for policy and, at face value,
seem to make a compelling case for the encouragement of competition.
The Two Theorems are easily demonstrated for a two-consumer exchange
economy by using the Edgeworth box diagram. The first step is to isolate the
Pareto-efficient allocations. Consider Figure 2.11 and the allocation at point
a. To show that a is not a Pareto-efficient allocation it is necessary to find an
alternative allocation which gives at least one of the consumers a higher utility
level and neither consumer a lower level. In this case, moving to the allocation
at point b raises the utility of both consumers when compared to a — we say
in such a case that b is Pareto-preferred to a. This establishes that a is not
Pareto-efficient. Although b improves upon a it is not Pareto-efficient either:
the allocation at c provides higher utility for both consumers than b.
The allocation at c is Pareto-efficient. Beginning at c any change in the
allocation must lower the utility of at least one of the consumers. The special
property of point c is that it lies at a point of tangency between the indifference
curves of the two consumers. As it is a point of tangency, moving away from
it must lead to a lower indifference curve for one of the consumers if not both.
Since the indifference curves are tangential, their gradients are equal so
1
2
= MRS1,2
.
MRS1,2
(2.18)
This equality ensures that the rate at which consumer 1 will want to exchange
good 1 for good 2 is equal to the rate at which consumer 2 will want to exchange
the two goods. It is this equality of the marginal valuations of the two consumers
at the tangency point that results in there being no further unexploited gains
and so makes c Pareto-efficient.
30
CHAPTER 2. EQUILIBRIUM AND EFFICIENCY
2
c
d
b
a
1
Figure 2.11: Pareto Efficiency
2
e
ω
1
Contract Curve
Figure 2.12: The First Theorem
The Pareto-efficient allocation at c is not unique. In fact, there are many
points of tangency between the two consumers’ indifference curves. A second
Pareto-efficient allocation is at point d in Figure 2.11. Taken together, all the
Pareto-efficient allocations form a locus in the Edgeworth box which is called
the contract curve. This is illustrated in Figure 2.12. With this construction it
is now possible to demonstrate the First Theorem.
A competitive equilibrium is given by a price line through the initial endowment point, ω, which is tangential to both indifference curves at the same
point. The common point of tangency results in consumer choices that lead to
the equilibrium levels of demand. Such an equilibrium is indicated by point e
in Figure 2.12. As the equilibrium is a point of tangency of indifference curves,
it must also be Pareto-efficient. For the Edgeworth box, this completes the
demonstration that a competitive equilibrium is Pareto-efficient.
2.4. EFFICIENCY OF COMPETITION
31
The alternative way of seeing this result is to recall that each consumer
maximizes utility at the point where their budget constraint is tangential to
the highest indifference curve. Using the M RS, this condition can be written
h
for consumer h as MRS1,2
= pp12 . The competitive assumption is that both
consumers react to the same set of prices so it follows that
1
MRS1,2
=
p1
2
= M RS1,2
.
p2
(2.19)
Comparing this condition with (2.18) provides an alternative demonstration
that the competitive equilibrium is Pareto-efficient. It also shows again the role
of prices in coordinating the independent decisions of different economic agents
to ensure efficiency.
This discussion can be summarized in the precise statement of the theorem.
Theorem 1 (The First Theorem of Welfare Economics) The allocation of commodities at a competitive equilibrium is Pareto-efficient.
This theorem can be formally proved by assuming that the competitive equilibrium is not Pareto-efficient and deriving a contradiction. Assuming the competitive equilibrium is not Pareto-efficient implies there is a feasible alternative
that is at least as good for all consumers and strictly better for at least one.
Now take the consumer who is made strictly better-off. Why did they not
choose the alternative consumption plan at the competitive equilibrium? The
answer has to be because it was more expensive than their choice at the competitive equilibrium and not affordable with their budget. Similarly, for all
other consumers the new allocation has to be at least as expensive as their
choice at the competitive equilibrium. (If it were cheaper, they could afford an
even better consumption plan which made them strictly better-off than at the
competitive equilibrium.) Summing across the consumers, the alternative allocation has to be strictly more expensive than the competitive allocation. But
the value of consumption at the competitive equilibrium must equal the value of
the endowment. Therefore the new allocation must have greater value than the
endowment which implies it cannot be feasible. This contradiction establishes
that the competitive equilibrium must be Pareto-efficient.
The theorem demonstrates that the competitive equilibrium is Pareto-efficient
but it is not the only Pareto-efficient allocation. Referring back to Figure 2.12,
any point on the contract curve is also Pareto-efficient since all are defined by a
tangency between indifference curves. The only special feature of e is that it is
the allocation reached through competitive trading from the initial endowment
point ω. If ω were different, then another Pareto-efficient allocation would be
achieved. In fact there is an infinity of Pareto-efficient allocations. Observing
these points motivates the Second Theorem of Welfare Economics.
The Second Theorem is concerned with whether any chosen Pareto-efficient
allocation can be made into a competitive equilibrium by choosing a suitable
location for the initial endowment. Expressed differently, can a competitive
economy be constructed which has a selected Pareto-efficient allocation as its
32
CHAPTER 2. EQUILIBRIUM AND EFFICIENCY
2
e'
ω
ω'
1
Figure 2.13: The Second Theorem
competitive equilibrium? In the Edgeworth box, this involves being able to
choose any point on the contract curve and turning it into a competitive equilibrium.
Using the Edgeworth Box diagram it can be seen that this is possible in
the exchange economy if the households’ indifference curves are convex. The
common tangent to the indifference curves at the Pareto-efficient allocation
provides the budget constraint that each consumer must face if they are to
afford the chosen point. The convexity ensures that, given this budget line,
the Pareto-efficient point will also be the optimal choice of the consumers. The
construction is completed by choosing a point on this budget line as the initial
endowment point. This process of constructing a competitive economy to obtain
a selected Pareto-efficient allocation is termed decentralization.
This process is illustrated in Figure 2.13 where the Pareto-efficient allocation
e is made a competitive equilibrium by selecting ω as the endowment point.
Starting from ω , trading by consumers will take the economy to its equilibrium
allocation e . This is the Pareto-efficient allocation that it was intended should
be reached. Note that if the endowments of the households are initially given by
ω and the equilibrium at e is to be decentralized, it is necessary to redistribute
the initial endowments of the consumers in order to begin from ω .
The construction described above can be given a formal statement as the
Second Theorem of Welfare Economics.
Theorem 2 (The Second Theorem of Welfare Economics) With convex preferences, any Pareto-efficient allocation can be made a competitive equilibrium.
The statement of the Second Theorem provides a conclusion but does not
describe the mechanism involved in the decentralization. The important step
in decentralizing a chosen Pareto-efficient allocation is placing the economy at
the correct starting point. For now it is sufficient to observe that behind the
Second Theorem lies a process of redistribution of initial wealth. How this can
2.4. EFFICIENCY OF COMPETITION
33
be achieved is discussed later. Furthermore, the Second Theorem determines a
set of prices that make the chosen allocation an equilibrium. These prices may
well be very different from those that would have been obtained in the absence
of the wealth redistribution.
2.4.4
Extension to Production
The extension of the Two Theorems to an economy with production is straightforward. The major effect of production is to make supply variable: it is now the
sum of the initial endowment plus the net outputs of the firms. In addition, a
consumer’s income includes the profit derived from their shareholdings in firms.
Section 2.4.1 has already demonstrated efficiency for the Robinson Crusoe
economy which included production. It was shown that the competitive equilibrium achieved the highest attainable indifference curve given the production
possibilities of the economy. Since the single consumer cannot be made better off
by any change, the equilibrium is Pareto-efficient and the First Theorem applies.
The Second Theorem is of limited interest with a single consumer since there
is only one Pareto-efficient allocation and this is attained by the competitive
economy.
When there is more than one consumer the proof of the First Theorem follows
the same lines as for the exchange economy. Given the equilibrium prices, each
consumer is maximizing utility so their marginal rate of substitution is equated
to the price ratio. This is true for all consumers and all goods, so
h
=
MRSi,j
pi
h
= M RSi,j
,
pj
(2.20)
for any pair of consumers h and h and any pair of goods i and j. This is termed
efficiency in consumption. In an economy with production this condition alone
is not sufficient to guarantee efficiency and it is also necessary to consider production. The profit-maximization decision of each firm ensures that it equates
its marginal rate of transformation between any two goods to the ratio of prices.
For any two firms m and m this give
m
MRTi,j
=
pi
m
= MRTi,j
,
pj
(2.21)
a condition that characterizes efficiency in production. The price ratio also
coordinates consumers and firms giving
h
m
= MRTi,j
,
MRSi,j
(2.22)
for any consumer and any firm for all pairs of goods. As for the RobinsonCrusoe economy, the interpretation of this condition is that it equates the relative marginal values to the relative marginal costs. Since (2.20) - (2.22) are the
conditions required for efficiency, this shows that the First Theorem extends to
the economy with production.
34
CHAPTER 2. EQUILIBRIUM AND EFFICIENCY
Z
x̂
W
Price
Line
x̂1
x̂ 2
Feasible
Set
Figure 2.14: Proof of the Second Theorem
The formal proof of this claim mirrors that for the exchange economy, except for the fact that the value of production must also be taken into account.
Given this, the basis of the argument remains that since the consumers chose
the competitive equilibrium quantities anything that is preferred must be more
expensive and hence can be shown not to be feasible.
The extension of the Second Theorem to include production is illustrated in
Figure 2.14. The set W describes the feasible output plans for the economy with
each point in W being the sum of a production plan and the initial endowment,
hence w = y+ω. Set Z describes the quantities of the two goods that would allow
a Pareto-improvement (a re-allocation that makes neither consumer worse-off
2 to
and makes one strictly better-off) over the allocation x
1 to consumer 1 and x
consumer 2. If W and Z are convex, which occurs when firms’ production sets
and preferences are convex, then a common tangent to W and Z can be found.
This would make x
an equilibrium. Individual income allocations, the sum of the
value of endowment plus profit income, can be placed anywhere on the budget
lines tangent to the indifference curves at the individual allocations x
1 and x
2
provided they sum to the total income of the economy. This decentralizes the
consumption allocation x
1 , x
2 .
Before proceeding further, it is worth emphasizing that the proof of the
Second Theorem requires more assumptions than the proof of the First, so there
may be situations in which the First Theorem is applicable but the Second is not.
The Second Theorem requires that a common tangent can be found which relies
on preferences and production sets being convex. A competitive equilibrium
can exist with some non-convexity in the production sets of the individual firms
or the preferences of the consumers, so the First Theorem will apply, but the
Second Theorem will not apply.
2.5. LUMP-SUM TAXATION
2.5
35
Lump-Sum Taxation
The discussion of the Second Theorem noted that it did not describe the mechanism through which the decentralization is achieved. Instead, it was implicit
in the statement of the theorem that the consumers would be given sufficient income to purchase the consumption plans forming the Pareto-efficient allocation.
Any practical value of the Second Theorem depends on being able to achieve
these required income levels. The way in which the theorem sees this as being
done is by making what are called lump-sum transfers between consumers.
A transfer is defined as lump-sum if no change in a consumer’s behavior can
affect the size of the transfer. For example, a consumer choosing to work less
hard or reducing the consumption of a commodity must not be able to affect the
size of the transfer. This differentiates a lump-sum transfer from other taxes,
such as income or commodity taxes, for which changes in behavior do affect the
value of the tax payment. Lump-sum transfers have a very special role in the
theoretical analysis of public economics because, as we will show, they are the
idealized redistributive instrument.
The lump-sum transfers envisaged by the Second Theorem involve quantities
of endowments and shares being transferred between consumers to ensure the
necessary income levels. Some consumers would gain from the transfers, others
would lose. Although the value of the transfer cannot be changed, lump-sum
transfers do affect consumers’ behavior since their incomes are either reduced
or increased by the transfers — the transfers have an income effect but do not
lead to a substitution effect between commodities. Without recourse to such
transfers, the decentralization of the selected allocation would not be possible.
The illustration of the Second Theorem in an exchange economy in Figure
2.15 makes clear the role and nature of lump-sum transfers. The initial endowment point is denoted ω and this is the starting point for the economy.
Assuming that the Pareto-efficient allocation at point e is to be decentralized,
the income levels have to be modified to achieve the new budget constraint. At
the initial point, the income level of h is p
ωh when evaluated at the equilibrium
prices p
. The value of the transfer to consumer h that is necessary to achieve
the new budget constraint is M h − p
ω h = p
x
h − p
ωh . One way of ensuring this
is to transfer a quantity x
11 of good 1 from consumer 1 to consumer 2. But any
transfer of commodities with the same value would work equally well.
There is a problem, though, if we attempt to interpret the model this literally.
For most people, income is earned almost entirely from the sale of labor so that
their endowment is simply lifetime labor supply. This makes it impossible to
transfer the endowment since one person’s labor cannot be given to another.
Responding to such difficulties leads to the reformulation of lump-sum transfers
in terms of lump-sum taxes. Suppose that the two consumers both sell their
entire endowments at prices p
. This generates incomes p
ω1 and p
ω2 for the two
consumers. Now make consumer 1 pay a tax of amount T 1 = p
x
11 and give
this tax revenue to consumer 2. Consumer 2 therefore pays a negative tax (or,
2
px
11 = −T 1 . The pair of taxes
in 1simpler
terms, receives a subsidy) of T = −
T , T 2 moves the budget constraint in exactly the same way as the lump-sum
36
CHAPTER 2. EQUILIBRIUM AND EFFICIENCY
2
e
ω
ω'
1
~
x11
Figure 2.15: A Lump-Sum Transfer
transfer of endowment. The pair of taxes and the transfer of endowment are
therefore economically equivalent and have the same effect upon the economy.
The taxes are also lump-sum since they are determined without reference to
either consumers’ behavior and their values cannot be affected by any change
in behavior.
Lump-sum taxes have a central role in public economics due to their efficiency in achieving distributional objectives. It should be clear from the discussion above that the economy’s total endowment is not reduced by the application
of the lump-sum taxes. This point applies to lump-sum taxes in general. As
households cannot affect the level of the tax by changing their behavior, lumpsum taxes do not lead to any distortions in choice. There are also no resources
lost due to the imposition of lump-sum taxes and so redistribution is achieved
with no efficiency cost. In short, if they can be employed in the manner described they are the perfect taxes.
2.6
Discussion of Assumptions
The description of the competitive economy introduced a number of assumptions
concerning the economic environment and how trade was conducted. These are
important since they bear directly upon the efficiency properties of competition.
The interpretation and limitation of these assumptions is now discussed. This
should help to provide a better context for evaluating the practical relevance of
the efficiency theorems.
The most fundamental assumption was that of competitive behavior. This
is the assumption that both consumers and firms view prices as fixed when
they make their decisions. The natural interpretation of this assumption is that
the individual economic agents are small relative to the total economy. When
they are small, they naturally have no consequence. This assumption rules out
any kind of market power such as monopolistic firms or trade unions in labor
2.6. DISCUSSION OF ASSUMPTIONS
37
markets.
Competitive behavior leads to the problem of who actually sets prices in
the economy. In the exchange model it is possible for equilibrium prices to be
achieved via a process of barter and negotiation between the trading parties.
However barter cannot be a credible explanation of price determination in an
advanced economic environment. One theoretical route out of this difficulty is
to assume the existence of a fictitious “Walrasian auctioneer” who literally calls
out prices until equilibrium is achieved. Only at this point trade is allowed to
take place. Obviously, this does not provide a credible explanation of reality.
Although there are other theoretical explanations of price setting, none is entirely consistent with the competitive assumption. How to integrate the two
remains an unsolved puzzle.
The second assumption was symmetry of information. In a complex world
there are many situations in which this does not apply. For instance, some
qualities of a product, such as reliability (I do not know when my computer will
next crash, but I expect it will be soon), are not immediately observable but are
discovered only through experience. When it comes to re-sale, this causes an
asymmetry of information between the existing owner and potential purchasers.
The same can be true in labor markets where workers may know more about
their attitudes to work and potential productivity than a prospective employer.
An asymmetry of information provides a poor basis for trade since the caution
of those transacting prevents the full gains from trade being realized.
When any of the assumptions underlying the competitive economy fail to be
met and as a consequence efficiency is not achieved, we say that there is market
failure. Situations of market failure are of interest to public economics since
they provide a potential role for government policy to enhance efficiency. In
fact, a large section of this book is devoted to a detailed analysis of the sources
of market failure and the scope for policy response.
As a final observation, notice that the focus has been upon positions of
equilibrium. Several explanations can be given for this emphasis. Historically,
economists viewed the economy as self-correcting so that, if it were ever away
from equilibrium, forces existed that move it back towards equilibrium. In the
long-run, equilibrium would then always be attained. Although such adjustment
can be justified in simple single-market contexts, both the practical experience of
sustained high levels of unemployment and the theoretical study of the stability
of the price adjustment process have shown that it is not generally justified. The
present justifications for focusing upon equilibrium are more pragmatic. The
analysis of a model must begin somewhere, and the equilibrium has much merit
as a starting point. In addition, even if the final focus is on disequilibrium, there
is much to be gained from comparing the properties of points of disequilibrium
to those of the equilibrium. Finally, no positions other than those of equilibrium
have any obvious claim to prominence.
38
CHAPTER 2. EQUILIBRIUM AND EFFICIENCY
2.7
Summary
This chapter described competitive economies and demonstrated the Two Theorems of Welfare Economics. To do this it was necessary to introduce the
concept of Pareto-efficiency. Whilst this was simply accepted in this chapter, it
will be considered very critically in Chapter 12. The Two Theorems characterize
the efficiency properties of the competitive economy and show how a selected
Pareto-efficient allocation can be decentralized. It was also shown how prices
are central to the achievement of efficiency through their role in coordinating
the choices of individual agents. The role of lump-sum transfers or taxes in
supporting the Second Theorem was highlighted. These transfers constitute the
ideal tax system since they cause no distortions in choice and have no resource
costs.
The subject matter of this chapter has very strong implications which are
investigated fully in later chapters. An understanding of them, and of their
limitations, is fundamental to appreciating many of the developments of public
economics. Since claims about the efficiency of the competition feature routinely
in economic debate it is important to subject it to the most careful scrutiny.
Further Reading
The two fundamental texts on the competitive economy are:
Arrow, K.J. and Hahn, F.H. (1971) General Competitive Analysis (Amsterdam: North-Holland),
Debreu, G. (1959) The Theory of Value (Yale: Yale University Press).
A textbook treatment can be found in:
Ellickson, B. (1993) Competitive Equilibrium: Theory and Applications (Cambridge: Cambridge University Press).
The competitive economy has frequently been used as a practical tool for
policy analysis. A survey of applications is in:
Shoven, J.B. and Whalley, J. (1992) Applying General Equilibrium Theory
(Cambridge: Cambridge University Press).
A historical survey of the development of the model is given in:
Duffie, D. and Sonnenschein, H. (1989) “Arrow and General Equilibrium
Theory”, Journal of Economic Literature, 27, 565 - 598.
Some questions concerning the foundations of the model are addressed in:
Koopmans, T.C. (1957) Three Essays on the State of Economic Science
(New York: McGraw-Hill).
The classic proof of the Two Theorems is in:
Arrow, K.J. (1951) “An extension of the basic theorems of welfare economics”, in J. Neyman (ed.), Proceedings of the Second Berkeley Symposium
on Mathematical Statistics and Probability, Berkeley: University of California
Press.
A formal analysis of lump-sum taxation can be found in:
Mirrlees, J.A. (1986) “The theory of optimal taxation” in K.J. Arrow and
M.D. Intrilligator (eds.), Handbook of Mathematical Economics (Amsterdam:
North-Holland).